Geokon Stressmeter 4300 Series, Инструкция по эксплуатации

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12
Appendix 1: Biaxial Stress Changes ______________________________
The relationship between the radial deformation of a borehole, U, and two principle stresses in the plane of a
borehole has been given by Hast (1958) and Merrill and Peterson (1961).
For Plane Stress:
U = d/E
r
[(
σ
1
+
σ
2
) + 2 (
σ
1
-
σ
2
) cos 2
θ
]
Where:
σ
1
and
σ
2
are the principle stresses in the plane of the borehole.
θ
is the angle measured counterclockwise from the direction of
σ
1
.
d is the diameter of the borehole.
E
r
is the Youngs modulus of the rock
If we assume that the stress measured across the stressmeter is proportional to the radial deformation that would
have occurred in this direction if the stressmeter had not been there, then the term d/E
r
can be replaced by one
reflecting the relationship between the rock modulus and the gage modulus. Hast (1958) has shown this to be
applicable for a uniaxial stressmeter. For the measurement of stress
σ
R
in any direction
θ
the following applies:
σ
R
= 1/3 (
σ
1
+
σ
2
) + 2/3 (
σ
1
-
σ
2
) cos 2
θ
(
θ
measured counterclockwise from
σ
1
)
Using this relationship and three uniaxial stress change measurements at 45
°
to each other, the secondary
principle stresses
σ
1
and
σ
2
and the angle
θ
are given by:
σ
1
= 3/2 a + ¾ b
σ
2
= 3/2 a – ¾ b
θ
= 1/2 sin
-1
((a –
σ
45
)/b)
where: a =
σ
0
+
σ
90
/2
b = [(
σ
45
– a)
2
+(
σ
0
– a)
2
]
1/2
To determine the
θ
angles, you have to determine what quadrant the angle lies in. The inequalities to do this are
as follows:
If
σ
45
≤
a and
σ
0
≥
90, then 0
≤ θ ≥
45
°
If
σ
45
≤
a and
σ
0
≤
. 90, then 45
° ≤ θ ≤
90
°
If
σ
45
≥
a and
σ
0
≤
90, then 90
° ≤ θ ≤
135
°
If
σ
45
≥
a and
σ
0
≥
90, then 135
° ≤ θ ≤
180
°
Note:
θ
is measured clockwise for
σ
0
(this is same as counter-clockwise for
σ
1
).